The existence of cyclic variability in the quality of combustion in spark-ignited, internal combustion engine has long been recognized. Such variations can be particularly severe for lean air-fuel mixtures, i.e., when the ratio of air to fuel is greater than that implied by chemical stoichiometry. The analysis of these variations is made difficult by the existence of several possible mechanisms which could act separately or in concert. One problem is the variations in the delivery of air and fuel into the cylinder. The effect of variations either in mass of fuel or its distribution would tend to be exacerbated under lean conditions, when the total mass of fuel is relatively smaller. It is recognized that the fluid dynamic effects during engine intake and exhaust strokes are dominant contributors in cyclic variations. The importance of the residual gas, both content and amount, has also been recognized and is generally regarded as the cause of the frequently observed alternating pattern of high and low work output cycles, although other mechanisms have been proposed. Investigators have considered cyclic variations from the standpoint of understanding the mechanism well enough to effect a reduction in the variation by imposing control. They found that significant correlation exists between consecutive firings of a particular cylinder, and that various relevant measurable quantities, such as indicated mean effective pressure, are subject to reasonable prediction one cycle in advance. Various means of imposing control, such as through changes of spark timing and fuel delivery have been considered.
Because the combustion process depends on several state variables and is nonlinear, it is a candidate for exhibiting the complex behavior called deterministic chaos, or just chaos for short. If chaotic behavior takes place in a system with many important state variables (e.g., more than ten), it is termed high-dimensional chaos. While high dimensional chaos is in principle deterministic, it is usually so complex that as a practical matter (at least with current understanding), it can only be treated with methods applicable to stochastic (random) systems. Hence to be of present practical importance, e.g., for better fundamental understanding or real-time control of a physical system, it is necessary for the identified chaotic behavior to be low -dimensional, (e.g., have a number of important state variables that is less than ten).
The possibility of chaotic behavior in spark ignited engines was suggested at least as far back as 1984 and has been a continuing source of investigation. Chew et al. claim to have identified chaotic behavior in a production internal combustion engine, though they are not rigorous in distinguishing their observations from stochastic behavior. Further, they do not discuss implications for practical application of their findings. Finney, Nguyen, and Daw (Japanese Combustion Symposium, Sendai, 1994) make use of chaotic time series analysis to analyze data from a one-cylinder engine, concluding that the variations observed are not consistent with purely random behavior and hence that short-term predictability might be possible.
In an attempt to explain how residual gas affects could lead to nonlinear deterministic coupling between cycles, Daw et al. developed a simple model for variations of fuel and air in an engine cylinder: EQU m(i+1)=m(i)*(1-CE)*F+(1-F)*MF+.delta.MF(i)! EQU a(i+1)=a(i)-R*CE*m(i)!*F+(1-F)*AF
where the main variables are defined as:
m(i)=mass of fuel before ith burn PA1 a(i)=mass of air before ith burn PA1 .delta.MF(i)=small change in mass of fresh fuel per cycle, dictated by control; PA1 MF=mass of fresh fuel per cycle PA1 AF=mass of fresh air fed per cycle; PA1 F=fraction cylinder gas remaining PA1 R=stoichiometric air-fuel ratio, .about.14.6; PA1 CE=combustion efficiency of ith burn.
constant, or slowly varying variables are:
parameters whose values are indicated by engine or fuel characteristics are:
and a key variable which may, for example, be a function of air-fuel ratio:
With no control imposed (.delta.MF(i)=0) and no stochastic perturbation of the parameters, this model exhibits unstable behavior in the form of period-doubling and chaotic behavior at lean conditions, depending on particular values of the variables and parameters and the functional form and characteristics of CE. Given simple functional forms for CE, the equations may be solved for fixed points in the variables, i.e., values where the behavior is at least marginally stable, and a control equation may be developed that will force the system to a fixed point and keep it there. For practical application, the functional form of CE is not known a priori and must be developed from heuristic arguments and experience. Nevertheless, it is expected from combustion physics that the functional form of CE includes a strong nonlinear dependence of combustion efficiency on the in-cylinder fuel and air content at the time of the burn.
The initial period-2 bifurcation of the uncontrolled model represents a condition where the fixed point becomes unstable due to the effect of the nonlinearity. The bifurcation is physically explained by considering that the residual mass fraction for a slow burn or partial misfire enhances the fuel-air ratio for the next burn. Similarly, a strong burn will leave no fuel in the residual gas, leading to the possibility of a leaner than average mixture and lower output for the next burn. Near stoichiometry, such small changes would have little impact, but as the strongly nonlinear lean combustion boundary is approached, small changes in cylinder inventory produce large consequences. When the alternating strong and weak burns occur, it is expected to appear as an anti-correlation in the time series of combustion indices such as heat release and IMEP.
Additionally, there is considerable uncertainty or noise associated with the combustion process. Such noise can be described in terms of stochastic (typically Gaussian) variations in the model parameters. The model nonlinearities amplify the effect of these stochastic variations, and the tendency to go into oscillations and chaos is increased. Although the presence of such noise complicates the cyclic variation patterns, their global features continue to be dominated by the characteristics of the unperturbed nonlinear system.
Because the effects of the nonlinear determinism continues to dominate even in the presence of noise, an adaptive control approach of the following form can be used to reduce cyclic variations: EQU .delta.MF(i)=Gain*(CE(i)-tgtCE(i)) EQU tgtCE(i+1)=tgtCE(i)+scalar*.delta.MF(i)
where: EQU tgtCE=desired or target CE
and Gain and scalar are parameters to be determined experimentally. This approach may also work when CE is not available but there is some quantity that is well correlated with it, such as heat release or acceleration: EQU .delta.MF(i)=Gain*(accel(i)-tgtaccel(i)) EQU tgtaccel(i+1)=tgtaccel(i)+scalar*.delta.MF(i)
where: EQU tgtaccel=desired or target accel.
In controlled test cell experiments with an eight-cylinder, 4.6L, 2-valve engine it has been demonstrated that the general patterns predicted by the above simple model are actually produced under lean fueling conditions (Daw et al.,). Specifically, dynamic combustion variations in standard indicators such as heat release and IMEP were monitored over several thousand cycles by recording and processing the in-cylinder pressure. Analyzing these data with techniques from chaotic time series analysis (e.g., time delay embedding and return maps), provided strong evidence that the combustion becomes unstable with increasingly lean operation via a period-2 bifurcation sequence that leads to alternating low- and high-power strokes. This bifurcation pattern is clearly visible in test cell measurements in spite of the stochastic parameter perturbations that are known to be occurring in the experiments. At extremely lean fueling it appears that the engine becomes fully chaotic, although this condition is so erratic it would not seem to be of interest for passenger automobiles. As predicted by the model, increases in the magnitude of the stochastic inputs tends to accelerate the onset of the bifurcations (i.e., they begin to occur at higher equivalence ratios). Comparisons of experimental return map patterns with model predictions show strong similarities that confirm the basic correctness of this model (see FIGS. 9-12, Daw et al.,).